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Theorem exbtwnzlemex 9388
Description: Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the  n  <_  A  \/  A  <  n hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition.

The proof starts by finding two integers which are less than and greater than  A. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the  n  <_  A  \/  A  <  n hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

Hypotheses
Ref Expression
exbtwnzlemex.a  |-  ( ph  ->  A  e.  RR )
exbtwnzlemex.tri  |-  ( (
ph  /\  n  e.  ZZ )  ->  ( n  <_  A  \/  A  <  n ) )
Assertion
Ref Expression
exbtwnzlemex  |-  ( ph  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Distinct variable groups:    A, n    x, A    ph, n
Allowed substitution hint:    ph( x)

Proof of Theorem exbtwnzlemex
Dummy variables  a  j  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exbtwnzlemex.a . . . 4  |-  ( ph  ->  A  e.  RR )
2 btwnz 8599 . . . 4  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j ) )
31, 2syl 14 . . 3  |-  ( ph  ->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j ) )
4 reeanv 2528 . . 3  |-  ( E. m  e.  ZZ  E. j  e.  ZZ  (
m  <  A  /\  A  <  j )  <->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j
) )
53, 4sylibr 132 . 2  |-  ( ph  ->  E. m  e.  ZZ  E. j  e.  ZZ  (
m  <  A  /\  A  <  j ) )
6 simplrl 502 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  ZZ )
76zred 8602 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  RR )
81ad2antrr 472 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  e.  RR )
9 simprl 498 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  A )
107, 8, 9ltled 7347 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <_  A )
11 simprr 499 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  j )
126zcnd 8603 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  CC )
13 simplrr 503 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  ZZ )
1413zcnd 8603 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  CC )
1512, 14pncan3d 7541 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  +  ( j  -  m ) )  =  j )
1611, 15breqtrrd 3831 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  ( m  +  ( j  -  m
) ) )
17 breq1 3808 . . . . . . . 8  |-  ( y  =  m  ->  (
y  <_  A  <->  m  <_  A ) )
18 oveq1 5570 . . . . . . . . 9  |-  ( y  =  m  ->  (
y  +  ( j  -  m ) )  =  ( m  +  ( j  -  m
) ) )
1918breq2d 3817 . . . . . . . 8  |-  ( y  =  m  ->  ( A  <  ( y  +  ( j  -  m
) )  <->  A  <  ( m  +  ( j  -  m ) ) ) )
2017, 19anbi12d 457 . . . . . . 7  |-  ( y  =  m  ->  (
( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) )  <-> 
( m  <_  A  /\  A  <  ( m  +  ( j  -  m ) ) ) ) )
2120rspcev 2710 . . . . . 6  |-  ( ( m  e.  ZZ  /\  ( m  <_  A  /\  A  <  ( m  +  ( j  -  m
) ) ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
226, 10, 16, 21syl12anc 1168 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
2313zred 8602 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  RR )
247, 8, 23, 9, 11lttrd 7354 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  j )
25 znnsub 8535 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  j  e.  ZZ )  ->  ( m  <  j  <->  ( j  -  m )  e.  NN ) )
2625ad2antlr 473 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  <  j  <->  ( j  -  m )  e.  NN ) )
2724, 26mpbid 145 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( j  -  m
)  e.  NN )
28 exbtwnzlemex.tri . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ZZ )  ->  ( n  <_  A  \/  A  <  n ) )
2928ralrimiva 2439 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  ZZ  ( n  <_  A  \/  A  <  n ) )
30 breq1 3808 . . . . . . . . . . 11  |-  ( n  =  a  ->  (
n  <_  A  <->  a  <_  A ) )
31 breq2 3809 . . . . . . . . . . 11  |-  ( n  =  a  ->  ( A  <  n  <->  A  <  a ) )
3230, 31orbi12d 740 . . . . . . . . . 10  |-  ( n  =  a  ->  (
( n  <_  A  \/  A  <  n )  <-> 
( a  <_  A  \/  A  <  a ) ) )
3332cbvralv 2582 . . . . . . . . 9  |-  ( A. n  e.  ZZ  (
n  <_  A  \/  A  <  n )  <->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3429, 33sylib 120 . . . . . . . 8  |-  ( ph  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3534ad2antrr 472 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3635r19.21bi 2454 . . . . . 6  |-  ( ( ( ( ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  j ) )  /\  a  e.  ZZ )  ->  ( a  <_  A  \/  A  <  a ) )
3727, 8, 36exbtwnzlemshrink 9387 . . . . 5  |-  ( ( ( ( ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  j ) )  /\  E. y  e.  ZZ  (
y  <_  A  /\  A  <  ( y  +  ( j  -  m
) ) ) )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
3822, 37mpdan 412 . . . 4  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
3938ex 113 . . 3  |-  ( (
ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( ( m  < 
A  /\  A  <  j )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
4039rexlimdvva 2489 . 2  |-  ( ph  ->  ( E. m  e.  ZZ  E. j  e.  ZZ  ( m  < 
A  /\  A  <  j )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
415, 40mpd 13 1  |-  ( ph  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    e. wcel 1434   A.wral 2353   E.wrex 2354   class class class wbr 3805  (class class class)co 5563   RRcr 7094   1c1 7096    + caddc 7098    < clt 7267    <_ cle 7268    - cmin 7398   NNcn 8158   ZZcz 8484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206  ax-arch 7209
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485
This theorem is referenced by:  qbtwnz  9390  apbtwnz  9408
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